[Martin Taylor 931220 11:30]

(Cliff Joslyn 931219 16:20)

My dynamics book defines an "attractor" as a set of points in a phase

space (e.g. magnitude vs. change of magnitude) such that there exists

am equilibrium point within the attractor, so that if you begin the

system anywhere in the attractor, it will approach the equilibrium in

the limit. The "basin of attraction" is then the largest attractor. So

for the ball in the bowl (with friction), each circular region with

center at the bottom of the bowl is an attractor, the bottom is the

(stable) equilibrium, and the entire bowl is the basin of attraction.

Two points. First of nomenclature: the "attractor" is, in at least some

dynamics books, taken as being the limit toward which orbits in a "basin

of attraction" lead. The limit may be a point, a cycle, or a fractal

structure, but in any case it is the end-point, if such exists, of orbits

in the phase space. The phase space may thus contain many basins of

attraction, which are separated by "repellors" (the boundaries between

two or more basins of attraction). Your book may use different terminology,

but that's what I have been using in writing of attractors and the like.

Secondly: The attractor exists in the phase space, not the physical space.

The centre of the bowl is not an attractor, nor even is it within a single

basin of attraction. The phase space for this situation has two (or three

if you allow the ball to bounce) spatial dimensions and two (or three)

momentum dimensions. The "centre of the bowl" defines a subspace of

constant spatial location extended through the two (or three) dimensional

momentum space. The attractor is at the centre of the bowl at zero

momentum in each dimension of phase space, not just at the centre of the

bowl in physical space.

There is no sense except that of everyday connotation in which an "attractor"

is associated with "an object that attracts." (This for some other posters,

not for Cliff).

Martin